Renormalization of Singular Stochastic PDEs via Paracontrolled Distributions and Regularity Structures

This monograph presents a rigorous comparative analysis of the two dominant frameworks for solving singular stochastic partial differential equations (SPDEs): the theory of Regularity Structures introduced by Hairer and the Paracontrolled Distribution method developed by Gubinelli, Imkeller, and Perkowski. We address the fundamental ill-posedness arising from non-linear operations on Schwartz distributions, specifically within the context of the Kardar-Parisi-Zhang (KPZ) equation and the dynamical 43 model. We explicate the algebraic machinery of the BPHZ renormalization group required to remove ultraviolet divergences, demonstrating how counterterms arise naturally in both settings. Furthermore, we elucidate the reconstruction theorem in Regularity Structures and its analytic counterpart in the commutator estimates of Paracontrolled Calculus, establishing a transfinite correspondence between the algebraic decorations of the former and the Fourier-analytic paraproducts of the latter. The study culminates in a unified view of stochastic quantization for subcritical regimes.